Math Problem Statement

Let f(x) = 1.5^(1.6x). Estimate the derivative f'(2.1) to within two decimal places by using a small enough interval.

Solution

To estimate the derivative f(2.1)f'(2.1) for the function f(x)=1.51.6xf(x) = 1.5^{1.6x}, we can use the definition of the derivative with a small interval. This approach is known as the difference quotient:

f(2.1)f(2.1+h)f(2.1)hf'(2.1) \approx \frac{f(2.1 + h) - f(2.1)}{h}

We need to choose a small value for hh (typically h=0.01h = 0.01 or smaller) to ensure an accurate estimate. Let's calculate it using h=0.01h = 0.01 and the function f(x)=1.51.6xf(x) = 1.5^{1.6x}.

First, we calculate f(2.11)f(2.11) and f(2.1)f(2.1):

f(2.11)=1.51.6×2.11,f(2.1)=1.51.6×2.1f(2.11) = 1.5^{1.6 \times 2.11}, \quad f(2.1) = 1.5^{1.6 \times 2.1}

Then apply the difference quotient formula:

f(2.1)f(2.11)f(2.1)0.01f'(2.1) \approx \frac{f(2.11) - f(2.1)}{0.01}

I'll compute this next.The estimated derivative f(2.1)f'(2.1) using the difference quotient method with h=0.01h = 0.01 is approximately 2.542.54.

Would you like a more detailed explanation of the steps, or do you have any further questions?

Here are five related questions to expand your understanding:

  1. How does the choice of hh affect the accuracy of the derivative estimate?
  2. What is the exact derivative of the function f(x)=1.51.6xf(x) = 1.5^{1.6x}?
  3. How would you compute the derivative at a different point, like f(1.5)f'(1.5)?
  4. What is the geometric interpretation of the derivative at x=2.1x = 2.1?
  5. How can you improve the precision of this estimate using smaller intervals?

Tip: Using a smaller hh in the difference quotient will generally increase the accuracy of your estimate. However, if hh becomes too small, numerical errors may arise.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Exponential Functions

Formulas

f'(x) ≈ (f(x+h) - f(x))/h
f(x) = 1.5^(1.6x)

Theorems

Difference Quotient Method for Numerical Derivatives

Suitable Grade Level

Undergraduate Calculus or Advanced High School